DIT411/TIN175, Artificial Intelligence Chapter 3: Classical search algorithms

CHAPTER 3: CLASSICAL SEARCHCHAPTER 3: CLASSICAL SEARCHALGORITHMSALGORITHMS

DIT411/TIN175, Artificial Intelligence

Peter Ljunglöf

19 January, 2018

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DEADLINE FOR FORMING GROUPSDEADLINE FOR FORMING GROUPS

Today is the deadline for forming groups.

if you have any problems, please talk to me in the breake.g., if you cannot contact one of your group membersor if you don’t have a group yet

Don’t forget to create a Github team and clone the Shrdlite repository

then email me with information about your group

Today we will decide your exact supervision time and your supervisor

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TABLE OF CONTENTSTABLE OF CONTENTSIntroduction (R&N 3.1–3.3)

Graphs and searchingExample problemsA generic searching algorithm

Uninformed search (R&N 3.4)Depth-first searchBreadth-first searchUniform-cost searchUniform-cost search

Heuristic search (R&N 3.5–3.6)Greedy best-first searchA* searchAdmissible and consistent heuristics

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INTRODUCTION (R&N 3.1–3.3)INTRODUCTION (R&N 3.1–3.3)GRAPHS AND SEARCHINGGRAPHS AND SEARCHING

EXAMPLE PROBLEMSEXAMPLE PROBLEMS

A GENERIC SEARCHING ALGORITHMA GENERIC SEARCHING ALGORITHM

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GRAPHS AND SEARCHINGGRAPHS AND SEARCHING

O�en we are not given an algorithm to solve a problem, but only a specification of a solution — we have to search for it.

A typical problem is when the agent is in one state, it has a set of deterministic actions it can carry out, and wants to get to a goal state.

Many AI problems can be abstracted into the problem of finding a path in a directed graph.

O�en there is more than one way to represent a problem as a graph.

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STATE-SPACE SEARCH: COMPLEXITY DIMENSIONSSTATE-SPACE SEARCH: COMPLEXITY DIMENSIONS

Observable? fullyDeterministic? deterministicEpisodic? episodicStatic? staticDiscrete? discreteN:o of agents single

Most complex problems (partly observable, stochastic, sequential) usualy have components using state-space search.

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DIRECTED GRAPHSDIRECTED GRAPHS

A graph consists of a set of nodes and a set of ordered pairs of nodes, called arcs or edges.

Node is a neighbor of if there is an arc from to . That is, if .

A path is a sequence of nodes such that .

The length of path is .

A solution is a path from a start node to a goal node, given a set of start nodes and goal nodes.

(Russel & Norvig sometimes call the graph nodes states).

N A

n2 n1 n1 n2

( , ) ∈ An1 n2

( , ,… , )n0 n1 nk ( , ) ∈ Ani−1 ni

( , ,… , )n0 n1 nk k

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EXAMPLE: TRAVEL IN ROMANIAEXAMPLE: TRAVEL IN ROMANIAWe want to drive from Arad to Bucharest in Romania

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EXAMPLE: GRID GAMEEXAMPLE: GRID GAMEGrid game: Rob needs to collect coins ,

without running out of fuel, and end up at location (1,1):

What is a good representation of the search states and the goal?

, , ,C1 C2 C3 C4

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EXAMPLE: VACUUM-CLEANING AGENTEXAMPLE: VACUUM-CLEANING AGENT

States [room A dirty?, room B dirty?, robot location]Initial state any stateActions le�, right, suck, do-nothingGoal test [false, false, –]

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EXAMPLE: THE 8-PUZZLEEXAMPLE: THE 8-PUZZLE

States a 3 x 3 matrix of integersInitial state any stateActions move the blank space: le�, right, up, downGoal test equal to the goal state

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EXAMPLE: THE 8-QUEENS PROBLEMEXAMPLE: THE 8-QUEENS PROBLEM

States any arrangement of 0 to 8 queens on the boardInitial state no queens on the boardActions add a queen to any empty squareGoal test 8 queens on the board, none attacked

This gives us possible paths to explore!64 × 63 ×⋯ × 57 ≈ 1.8 × 1014

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EXAMPLE: THE 8-QUEENS PROBLEM (ALTERNATIVE)EXAMPLE: THE 8-QUEENS PROBLEM (ALTERNATIVE)

States one queen per column in le�most columns, none attackedInitial state no queens on the boardActions add a queen to a square in the le�most empty column, make sure that no queen is attackedGoal test 8 queens on the board, none attacked

Using this formulation, we have only 2,057 paths!

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EXAMPLE: KNUTH’S CONJECTUREEXAMPLE: KNUTH’S CONJECTUREDonald Knuth conjectured that all positive integers can be obtained by starting with

the number 4 and applying some combination of the factorial, square root, and floor.

States algebraic numbers Initial state 4Actions apply factorial, square root, or floor operationGoal test a given positive integer (e.g., 5)

= 5

⎢

⎣⎢⎢ (4!)!‾ ‾‾‾√‾ ‾‾‾‾‾‾√

‾ ‾‾‾‾‾‾‾‾√‾ ‾‾‾‾‾‾‾‾‾‾‾

√‾ ‾‾‾‾‾‾‾‾‾‾‾‾‾

⎷

⎥

⎦⎥⎥

(1, 2.5, 9, , 1.23 ⋅ , ,…)2‾√ 10456 2‾√‾ ‾‾√

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EXAMPLE: ROBOTIC ASSEMBLYEXAMPLE: ROBOTIC ASSEMBLY

States real-valued coordinates of robot joint angles parts of the object to be assembledActions continuous motions of robot jointsGoal test complete assembly of the object

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HOW DO WE SEARCH IN A GRAPH?HOW DO WE SEARCH IN A GRAPH?

A generic search algorithm:

Given a graph, start nodes, and a goal description, incrementally explore paths from the start nodes.

Maintain a frontier of nodes that are to be explored.

As search proceeds, the frontier expands into the unexplored nodes until a goal node is encountered.

The way in which the frontier is expanded defines the search strategy.

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ILLUSTRATION OF GENERIC SEARCHILLUSTRATION OF GENERIC SEARCH

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A GENERIC TREE SEARCH ALGORITHMA GENERIC TREE SEARCH ALGORITHMTree search: Don’t check if nodes are visited multiple times

 function Search(graph, initialState, goalState):

initialise frontier using the initialState while frontier is not empty:

select and remove node from frontierif node.state is a goalState then return node for each child in ExpandChildNodes(node, graph):

add child to frontier return failure

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USING TREE SEARCH ON A GRAPHUSING TREE SEARCH ON A GRAPH

explored nodes might be revisitedfrontier nodes might be duplicated

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TURNING TREE SEARCH INTO GRAPH SEARCHTURNING TREE SEARCH INTO GRAPH SEARCHGraph search: Keep track of visited nodes

 function Search(graph, initialState, goalState):

initialise frontier using the initialStateinitialise exploredSet to the empty setwhile frontier is not empty:

select and remove node from frontierif node.state is a goalState then return nodeadd node to exploredSetfor each child in ExpandChildNodes(node, graph):

add child to frontier if child is not in frontier or exploredSetreturn failure

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TREE SEARCH VS. GRAPH SEARCHTREE SEARCH VS. GRAPH SEARCH

Tree search

Pro: uses less memoryCon: might visit the same node several times

Graph search

Pro: only visits nodes at most onceCon: uses more memory

Note: The pseudocode in these slides (and the course book) is not the only possible! E.g., Wikipedia uses a different variant.

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GRAPH NODES VS. SEARCH NODESGRAPH NODES VS. SEARCH NODESSearch nodes are not the same as graph nodes!Search nodes should contain more information:

the corresponding graph node (called state in R&N)the total path cost from the start nodethe estimated (heuristic) cost to the goalenough information to be able to calculate the final path

 procedure ExpandChildNodes(parent, graph):

for each (action, child, edgecost) in graph.successors(parent.state):yield new SearchNode(child,

            …total cost so far…,            …estimated cost to goal…,            …information for calculating final path…)

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UNINFORMED SEARCH (R&N 3.4)UNINFORMED SEARCH (R&N 3.4)DEPTH-FIRST SEARCHDEPTH-FIRST SEARCH

BREADTH-FIRST SEARCHBREADTH-FIRST SEARCH

UNIFORM-COST SEARCHUNIFORM-COST SEARCH

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QUESTION TIME: DEPTH-FIRST SEARCHQUESTION TIME: DEPTH-FIRST SEARCHWhich shaded goal will a depth-first search find first?

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QUESTION TIME: BREADTH-FIRST SEARCHQUESTION TIME: BREADTH-FIRST SEARCHWhich shaded goal will a breadth-first search find first?

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DEPTH-FIRST SEARCHDEPTH-FIRST SEARCH

Depth-first search treats the frontier as a stack.

It always selects one of the last elements added to the frontier.

If the list of nodes on the frontier is , then:

is selected (and removed).Nodes that extend are added to the front of the stack (in front of ).

is only selected when all nodes from have been explored.

[ , , ,…]p1 p2 p3

p1

p1 p2

p2 p1

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ILLUSTRATIVE GRAPH: DEPTH-FIRST SEARCHILLUSTRATIVE GRAPH: DEPTH-FIRST SEARCH

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COMPLEXITY OF DEPTH-FIRST SEARCHCOMPLEXITY OF DEPTH-FIRST SEARCH

Does DFS guarantee to find the path with fewest arcs?

What happens on infinite graphs or on graphs with cycles if there is a solution?

What is the time complexity as a function of the path length?

What is the space complexity as a function of the path length?

How does the goal affect the search?

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BREADTH-FIRST SEARCHBREADTH-FIRST SEARCH

Breadth-first search treats the frontier as a queue.

It always selects one of the earliest elements added to the frontier.

If the list of paths on the frontier is , then:

is selected (and removed).Its neighbors are added to the end of the queue, a�er .

is selected next.

[ , ,… , ]p1 p2 pr

p1

pr

p2

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ILLUSTRATIVE GRAPH: BREADTH-FIRST SEARCHILLUSTRATIVE GRAPH: BREADTH-FIRST SEARCH

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COMPLEXITY OF BREADTH-FIRST SEARCHCOMPLEXITY OF BREADTH-FIRST SEARCH

Does BFS guarantee to find the path with fewest arcs?

What happens on infinite graphs or on graphs with cycles if there is a solution?

What is the time complexity as a function of the path length?

What is the space complexity as a function of the path length?

How does the goal affect the search?

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UNIFORM-COST SEARCHUNIFORM-COST SEARCHWeighted graphs:

Sometimes there are costs associated with arcs. The cost of a path is the sum of the costs of its arcs.

An optimal solution is one with minimum cost.

Uniform-cost search (aka Lowest-cost-first search):Uniform-cost search selects a path on the frontier with the lowest cost.The frontier is a priority queue ordered by path cost.It finds a least-cost path to a goal node — i.e., uniform-cost search is optimalWhen arc costs are equal breadth-first search.

cost( ,… , ) = ( , )n0 nk ∑i=1

k

∣∣ ni−1 ni ∣∣

⇒

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HEURISTIC SEARCH (R&N 3.5–3.6)HEURISTIC SEARCH (R&N 3.5–3.6)GREEDY BEST-FIRST SEARCHGREEDY BEST-FIRST SEARCH

A* SEARCHA* SEARCH

ADMISSIBLE AND CONSISTENT HEURISTICSADMISSIBLE AND CONSISTENT HEURISTICS

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HEURISTIC SEARCHHEURISTIC SEARCHPrevious methods don’t use the goal to select a path to explore.

Main idea: don’t ignore the goal when selecting paths.

O�en there is extra knowledge that can guide the search: heuristics.

is an estimate of the cost of the shortest path from node to a goal node.

needs to be efficient to compute.

is an underestimate if there is no path from to a goal with cost less than .

An admissible heuristic is a nonnegative underestimating heuristic function:

h(n) n

h(n)

h(n) n

h(n)

0 ≤ h(n) ≤ cost(best path from n to goal)

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EXAMPLE HEURISTIC FUNCTIONSEXAMPLE HEURISTIC FUNCTIONS

Here are some example heuristic functions:

If the nodes are points on a Euclidean plane and the cost is the distance, can be the straight-line distance (SLD) from n to the closest goal.

If the nodes are locations and cost is time, we can use the distance to a goal divided by the maximum speed, (or the average speed, , which makes it non-admissible).

If the graph is a 2D grid maze, then we can use the Manhattan distance.

If the goal is to collect all of the coins and not run out of fuel, we can use an estimate of how many steps it will take to collect the coins and return to goal position, without caring about the fuel consumption.

A heuristic function can be found by solving a simpler (less constrained) version of the problem.

h(n)

h(n) = d(n)/vmax

h(n) = d(n)/vavg

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EXAMPLE HEURISTIC: ROMANIA DISTANCESEXAMPLE HEURISTIC: ROMANIA DISTANCES

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GREEDY BEST-FIRST SEARCHGREEDY BEST-FIRST SEARCH

Main idea: select the path whose end is closest to a goal according to the heuristic function.

Best-first search selects a path on the frontier with minimal -value.

It treats the frontier as a priority queue ordered by .

h

h

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GREEDY SEARCH EXAMPLE: ROMANIAGREEDY SEARCH EXAMPLE: ROMANIA

This is not the shortest path!

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GREEDY SEARCH IS NOT OPTIMALGREEDY SEARCH IS NOT OPTIMALGreedy search returns the path: Arad–Sibiu–Fagaras–Bucharest (450km)The optimal path is: Arad–Sibiu–Rimnicu–Pitesti–Bucharest (418km)

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BEST-FIRST SEARCH AND INFINITE LOOPSBEST-FIRST SEARCH AND INFINITE LOOPS

Best-first search might fall into an infinite loop!

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COMPLEXITY OF BEST-FIRST SEARCHCOMPLEXITY OF BEST-FIRST SEARCH

Does best-first search guarantee to find the path with fewest arcs?

What happens on infinite graphs or on graphs with cycles if there is a solution?

What is the time complexity as a function of the path length?

What is the space complexity as a function of the path length?

How does the goal affect the search?

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A* SEARCHA* SEARCH

A* search uses both path cost and heuristic values.

is the cost of path .

estimates the cost from the end node of to a goal.

, estimates the total path cost of going from the start node, via path to a goal:

cost(p) p

h(p) p

f (p) = cost(p) + h(p)

p

nstart  − →−−−path p

cost(p)

   goal− →−−−−estimate

h(p)

f (p)

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A* SEARCHA* SEARCH

A* is a mix of uniform-cost search and best-first search.

It treats the frontier as a priority queue ordered by .

It always selects the node on the frontier with the lowest estimated distance from the start to a goal node constrained to go via that node.

f (p)

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COMPLEXITY OF A* SEARCHCOMPLEXITY OF A* SEARCH

Does A* search guarantee to find the path with fewest arcs?

What happens on infinite graphs or on graphs with cycles if there is a solution?

What is the time complexity as a function of the path length?

What is the space complexity as a function of the path length?

How does the goal affect the search?

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A* SEARCH EXAMPLE: ROMANIAA* SEARCH EXAMPLE: ROMANIA

A* guarantees that this is the shortest path!

Note that we didn’t stop when we added Bucharest to the frontier. Instead, we stopped when we removed Bucharest from the frontier!

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A* SEARCH IS OPTIMALA* SEARCH IS OPTIMALThe optimal path is: Arad–Sibiu–Rimnicu–Pitesti–Bucharest (418km)

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A* ALWAYS FINDS A SOLUTIONA* ALWAYS FINDS A SOLUTION

A* will always find a solution if there is one, because:

The frontier always contains the initial part of a path to a goal, before that goal is selected.

A* halts, because the costs of the paths on the frontier keeps increasing, and will eventually exceed any finite number.

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ADMISSIBILITY (OPTIMALITY) OF ADMISSIBILITY (OPTIMALITY) OF A*A*

If there is a solution, A* always finds an optimal one first, provided that:

the branching factor is finite,

arc costs are bounded above zero (i.e., there is some such that all of the arc costs are greater than ), and

is nonnegative and an underestimate of the cost of the shortest path from to a goal node.

These requirements ensure that keeps increasing.

ϵ > 0

ϵ

h(n)

n

f

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WHY IS A* OPTIMAL?WHY IS A* OPTIMAL?

The values in A* are increasing, therefore:

first A* expands all nodes with

then A* expands all nodes with

finally A* expands all nodes with

A* will not expand any nodes with , where is the cost of an optimal solution.

(Note: all this assumes that the heuristics is admissible)

f

f (n) < C

f (n) = C

f (n) > C

f (n) > C∗C∗

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ILLUSTRATION: WHY IS A* OPTIMAL?ILLUSTRATION: WHY IS A* OPTIMAL?

A* gradually adds “ -contours” of nodes (cf. BFS adds layers)

Contour has all nodes with , where

f

i f = fi <fi fi+1

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QUESTION TIME: HEURISTICS FOR THE 8 PUZZLEQUESTION TIME: HEURISTICS FOR THE 8 PUZZLE = number of misplaced tiles = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

 =  8  =  3+1+2+2+2+3+3+2 = 18

(n)h1

(n)h2

(StartState)h1

(StartState)h2

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DOMINATING HEURISTICSDOMINATING HEURISTICS

If (admissible) for all , then dominates and is better for search.

Typical search costs (for 8-puzzle):

depth = 14 DFS  ≈  3,000,000 nodes A*( ) = 539 nodes A*( ) = 113 nodes

depth = 24 DFS  ≈  54,000,000,000 nodes A*( ) = 39,135 nodes A*( ) = 1,641 nodes

Given any admissible heuristics , , the maximum heuristics is also admissible and dominates both:

(n) ≥ (n)h2 h1 n

h2 h1

h1

h2

h1

h2

ha hb h(n)

h(n) = max( (n), (n))ha hb

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HEURISTICS FROM A RELAXED PROBLEMHEURISTICS FROM A RELAXED PROBLEM

Admissible heuristics can be derived from the exact solution cost of a relaxed problem:

If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then gives the shortest solution

If the rules are relaxed so that a tile can move to any adjacent square, then gives the shortest solution

Key point: the optimal solution cost of a relaxed problem is never greater than the optimal solution cost of the real problem

(n)h1

(n)h2

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SUMMARY OF TREE SEARCH STRATEGIESSUMMARY OF TREE SEARCH STRATEGIESSearch strategy

Frontierselection

Halts ifsolution?

Halts if nosolution?

Spaceusage

Depth first Last node added No No LinearBreadth first First node added Yes No Exp

Best first Global min No No Exp

Uniform cost Minimal Yes No Exp

A* Minimal Yes No Exp 

Halts if: If there is a path to a goal, it can find one, even on infinite graphs.Halts if no: Even if there is no solution, it will halt on a finite graph (with cycles).Space: Space complexity as a function of the length of the current path.

h(p)

cost(p)

f (p)

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EXAMPLE DEMOEXAMPLE DEMO

Here is an example demo of several different search algorithms, including A*. And you can play with different heuristics:

Note that this demo is tailor-made for planar grids, which is a special case of all possible search graphs.

(e.g., the Shrdlite graph will not be a planar grid)

http://qiao.github.io/PathFinding.js/visual/

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